Dont add a fish yet, the nitrites won't hurt them, but you want your whole cycle going so you don't get an ammonia spike (that hurts a lot). Also damsels are pretty fierce, it's usually better to add them last so they dont take over the whole tank. Quarantine is in a separate tank, to get the fish healthy before it goes in your DT.
They are all nitrogen. Just diff shapes of it. The last bacteria eats nitrates, but they can take months to get going so you don't need to wait for that to stock the tank. Just extra waterchanges in the beginning, like 10% per week is good.
Most of the bacteria is on the rocks and sand, the skimmer cant get it. It can't hurt but it might waste electricity.
umm... How is that diff from what I said?
But you are thinking a little too linearly. There are lots of processes occurring in our tanks that shift and alter nitrogen. For example: anammox, DNRA, AOA, as well as cyano fixing nitrogen gas, algae uptaking ammonia directly, zoox and diatoms consuming nitrite, and many many more.
It's not *just* a step by step, but OP probs won't see their nitrites go until the the denitrifiers have had enough time with the ammonia to get their numbers up. Like I said.